## TFY4210 Kvanteteori for mangepartikkelsystem

### Våren 2011

**Forelesar:**

Professor Jens O. Andersen (jensoa@ntnu.no)
,
Kontor: E5-145

**Innhald:**

Lagrange's equations for point particles, Lorentz transformations.
Classical field theories: Lagrange's equations, symmetries and conserved
quantities. Klein-Gordon and Dirac equation in external fields.
Field quantization of relativistic and nonrelativistic field theories.
Renormalization theory: cutoff and dimensional regularization.
First-order correction to ground-state energy and propagator.
Goldstone's theorem.
Dilute Bose gas and imperfect Fermi gas.
Dense electron gas.

**Forelesningar:**

**OBS:**
Mandag 08.15-10.00 i E5-103
og torsdag 14.15-16.00 i E5-103.
Eg har flytt rom fordi E5-103 og tavla er større.

Første forelesning mandag 10. januar.
Siste forelesning torsdag 14. april.

VIKTIG: Du er velkomen til å stikke innom viss du har
spørsmål eller lurer på noko.

**Øvingar:**

As part of the requirements, students will do the exercises in class.

**Spørretime**

Torsdag 14. April kl. 15.15-16.00 i E5-103.

**Eksamen:**

Tirsdag 24. mai 09.00-13.00.

Skisse
løysingsframlegg

Unfortunately, a couple of typos slipped through - however, none of importance.
I apologize for this and the numerous typos during the course.

Grades expected Tuesday May 31.

**Hjelpemiddel:**

Godkjend kalkulator

Rottmann: Matematisk Formelsamling

Rottmann: Matematische Formelsammlung

Schaum's Outline Series: Mathematical Handbook of Formulas and Tables

**Lærebok:**

A. L. Fetter and J. D. Walecka: Quantum Theory of Many-Particle Systems,
Dover Publications, 2003 (paperback). P. C. Hemmer: kvantemekanikk, Tapir
2005. Lecture notes av JOA: will appear below.
.
Please point out typos and send suggestions.

Lecture notes by Jan Myrheim and "an introduction to quantum field theory"
by M. E. Peskin and D. V. Schroeder (Addison Wesley) are recommended too.
**Course material:**

Chapters 1-10+appendix on scattering theory for NR Bose gases.

Chapter 10 and Appendix are
NOT
part of the curriculum this, but can be read optionally
by those of you who are interested in the scattering length a.

Chapter 17 in Hemmer.

Pages 64-72 (until Lehmann representation) in F+W.
Pages 21-28, 314-319 in Fetter and Walecka.

All exercises in class.

**Referansegrupppe:**

Eskil Aursand (aursand@stud.ntnu.no)

Aasmund Ervik (asmunder@stud.ntnu.no)

Sandra Hamann (hamann@stud.ntnu.no)

**Oppsummering:**

**Uke 2:**
Classical mechanics: Generalized coordinates and momenta. cyclic coordinates
and conserved quantities.
Newton's, Lagrange's and Hamilton's equations.
Calculus of variations: Extremizing functionals. Principle of least action and
the Euler-Lagrange equations. Lorentz transformations (boosts, rotations, and
translations). Contravariant and covariant vectors. Metric tensor and
scalar product. Metric in Minkowski space. Differential operators.

**Uke 3:**
Tensor of various types. Classical field theory: scalar and vector fields.
Lagrangian density and Euler-Lagrange equation for fields.
(Non)uniquesness of L. Discrete and continuous symmetries.
Noether's theorem, continuity equation and conserved quantities.
Global phase invariance and charge conservation. Space-time translations
and conserved energy and momentum.

**Uke 4:**
Complex scalar field and invariance under local phase transformations.
Introduction of gauge field and covariant derivative. Gauge transformations
Maxwell field,
field strength tensor, and its Lagrangian.
Nonrelativistic limit of field theories for scalar fields
(Schroedinger theory). Klein-Gordon equation in external electromagnetic
fields. Solution to the Coulomb problem (exact spectrum) and nonrelativistic
limit.

**Uke 5:**
Leading correction to NR-spectrum. Dirac equation and alpha-matrices.
Gamma-matrices and representation independence.
Lagrangian and Hamiltonian. Global phase symmetry and current conservation.

**Uke 6:**
Orbital angular momentum, spin, and conservation of J=L+S.
Free-particle solutions and spinors. Large and small components of
wavefunctions. Dirac equation in external fields. NR limit and relativistic
corrections. Exact Dirac spectrum. Harmonic oscillator.

**Uke 7:**
**
Lecture Monday 14/2-2011 cancelled. Exercise 4.3.2 is given as
homework as compensation.
**
Solution.
Harmonic oscillator and its algebra. Solution to the KG-equation and
Fourier expansion of fields. Quantization of free
fields by promoting Fourier coefficients to operators.
Hamiltonian and infinite vacuum (zero-point) energy.

**Uke 8:**
Qauntization of nonrelativistic field theories. Sum vs integral.
Fermions, negative energy solutions, and anticommutator.
Propagator in coordinate - and momentum space.

**Uke 9:**
Propagator as Greens' functions. Coulomb potential as real-space
propagator for Coulomb for static charge problem.
Self-interacting scalar theory. Perturbation theory and first-order correction
to the mass. Asymptotic series and g-2.
Vacuum fluctuations and virtual particles.
Cutoff field theories, bare and physical mass. Planck scale and
vacuum energy.
First-order correction to the vacuum energy.

**Uke 10:**
Vacuum energy to first order. Dimensional regularization and analytic
continuation in dimension d.
Mass correction and vacuum energy revisited using dimreg. Renormalization
and counterterms. Energy density and Helmholtz free energy density.

**Uke 11:**
Weakly interacting Bose gas:
Bogoliubov transformation and diagonalization of Hamiltonian, Bogoliubov
spectrum and quasiparticles. Gas parameter.
Ground state energy and coupling renormalization. Total density and
depletion of condensate due to interactions.

**Uke 12:**
Weakly interacting
Bose gas and spontaneous symmetry breaking of symmetries.
Effective potential.
Goldstone modes.
Propagator for free Fermi gas at finite density.
Density and energy density in terms of free propagator.
Interacting Fermi gas.

**Uke 13:**
First-order correction to the energy of an dense Fermi gas at T=0.
Effective potential for harmonic oscillator and ground-state energy.
Effective potential for Bose gas and renormalization. Photon propagator

**Uke 14:**
Degenerate electron gas: Background selfinteraction, electron-background
interaction. Regularization of infrared divergences by a photon mass.
Electron-electron interaction: cancellation of IR divergences and
removal of term from normal-ordering.

**Uke 15:**
Lecture Monday cancelled. Questions Thursday.

**Exercise set:**

**Uke 3:**
1.5.3, 2.5.4, and 2.5.5 (Ervik)
Løysingsframlegg

**Uke 4:**

**Uke 5:**
3.6.1 Monday (Aursand)
Løysingsframlegg
and
3.6.2 (part one) and 3.6.3 Thursday (Stige)
Løysingsframlegg

**Uke 6:**
3.6.4
(Glesaaen)
Løysingsframlegg

**Uke 7:**

**Uke 8:**
5.6.1 and 5.6.4 (Strymke)
Løysingsframlegg
.
4.3.1, 5.6.2 og 5.6.3.
(Garberg)
Løysingsframlegg

**Uke 9:**
5.6.5-5.6.7 (Bauer).
Løysingsframlegg

**Uke 10:**

**Uke 11:**
5.6.8 and 6.4.1. (Hamann).
Løysingsframlegg

**Uke 12:**
7.4.2 (Berge).
Løysingsframlegg

**Uke 13:**
7.4.1 (Ellingsen)
Løysingsframlegg

**Uke 14:**
Problem 3 Exam spring 2010.
(Yin)
Solution.

**Uke 15:**
Problem 9.3.1. (1-3) (Grzesiak)
.
**
**

Sist oppdatert 26. mai 2011 av Jens O. Andersen